CONTINUUM MECHANICS for ENGINEERS

This equation is clearly indeterminate for v = 0.5. However, show that
in this case Hooke’s law and the equilibrium equations yield the result
1
3 σ jj,i
Gu i,jj + + ρb i = 0
6.21 Let the displacement field be given in terms of some vector q i by the
equation
u
i
( ) −
21−vq
q
=
G
i,jj j,ji
Show that the Navier equation (Eq 6.4-7) is satisfied providing b i ≡ 0
and q i is bi-harmonic so that q i,jjkk = 0. If q 1 = x 2/r and q 2 = –x 1/r where
r 2 = x i x i, determine the resulting stress field.
Answer: σ11 = –σ22 = 6Qx1x2/r5 ; σ33 = 0; σ12 = 3QG /r5 2 2
x − x
σ13 = –σ31 = 3Qx2x3/r5 , σ23 = 0; where Q = 4(1 – v)/G
6.22 If body forces are zero, show that the elastodynamic Navier equation
(Eq 6.4-11) will be satisfied by the displacement field
u i = φ ,i + ε ijk ψ k,j
provided the potential functions φ and ψk satisfy the three-dimensional
wave equation.
6.23 Show that, for plane stress, Hooke’s law Eq 6.5-5 and Eq 6.5-6 may
be expressed in terms of the Lamé constants λ and µ by
ε
µ σ
λ
λ µ δσ
1 ⎛
⎞
ij = ⎜ ij −
ij kk⎟
2 ⎝ 3 + 2 ⎠
ε
33
λ
µ λ µ σ
=−
2 3 + 2
( ) ii
(i, j, k = 1, 2)
(i = 1, 2)
6.24 For the case of plane stress, let the stress components be defined in
terms of the function φ = φ(x 1, x 2), known as the Airy stress function,
by the relationships,
σ 12 = φ ,22, σ 22 = φ ,11, σ 12 = – φ ,12
( 2 1 )
Show that φ must satisfy the biharmonic equation 4 φ = 0 and that,
in the absence of body forces, the equilibrium equations are satisfied
3 2 5
identically by these stress components. If φ = Ax x −
Bx where A
1
2
1